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Unformatted text preview: Momentum vs. Wavevector • Instead of momentum, it is often convenient to use wavevector states: h P K = k k k K = ) ( k k k k ′ − = ′ δ π 2 x k i e k x = 1 = ∫ + ∞ ∞ − k k dk [ ] i K X = , [ ] , = K P k k k k k k p p p p ′ = ′ − = ′ − = ′ − = ′ h h h h 1 ) ( 1 ) ( ) ( δ δ δ ∴ k = h p p = h k Wavevector definition: Important commutators: Eigenvalue equation: Relation to momentum eigenstates: k p p c k h = = Normalization Closure Wavefunction Wavefunction in Kspace • What is the wavefunction in Kbasis? ) ( ) ( ) , ( 2 2 2 2 k k e k k e t k t k m t k i m t k i − = = = − − δ ψ ψ h h k k ) , ( t k ψ ) ( k = ψ Let 2 2 ) ( k e t t M k i h − = ψ Then: • Quantum mechanical kineticenergy eigenstate: – Only Global phase is changing in time – Global phase can’t be observed • Is anything actually moving? – The spatial phasepattern is moving, but at: – This is called the `phase velocity’ • Not related to particle velocity in classical limit • Classical freeparticle motion: • How can we have a `classical limit’ of QM if nothing moves at v=p /M? Wavefunction in Xspace ) ( ) ( p t p t v x t x = + = 2 2 ) ( p e t t M p i h − = ψ h h π ψ 2 ) , ( 2 − = t M p x p i e t x M p v 2 = h π ψ 2 1 ) , ( 2 = t x `Motion ’ in QM • The answer is that in QM motion is in interference effect • Consider a quantumsuperposition of two planewaves: • The interference pattern can move!...
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This note was uploaded on 12/21/2011 for the course PHY 851 taught by Professor Moore,m during the Fall '08 term at Michigan State University.
 Fall '08
 Moore,M
 mechanics, Momentum

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